Proof, series from k=1 to n of cos(2*pi*k/(2n+1)) equals -1/2

I have worked out a proof for relationships of complex numbers, and part of my proof requires me to prove this...

-1/2 = series from k=1 to n of cos(2*pi*k/(2n+1))

that is, for any positive integer n

I've wrote a program that evaluated this for an extended amount of integers n, in every case the series was equal to -1/2

However, I need a formal proof for this problem.

Another proof that I have to solve is this

0 = series from k=1 to n-1 of cos(pi*k/n)

that is, for any positive integer n greater than 1

this is easier to deduce, given that the values in the series will cancel each other out, or else be equivalent to 0

that being said, I'm looking for a formal proof for this series to

Any ideas?

Zander